arXiv subjects

Tamer A. Zaki

Publications and source records attributed to Tamer A. Zaki.

41 records · Page 3Linked to original sources

arXiv2018

Nonlinearly most unstable route to high-speed boundary-layer turbulence

Laminar-to-turbulence transition in zero-pressure-gradient boundary layer at Mach 4.5 is studied using direct numerical simulations. For a given level of total disturbance energy, the inflow spectra was designed to correspond to the nonlinearly most dangerous condition that leads to the earliest possible transition Reynolds number. The synthesis of the inlet disturbance is formulated as a constrained optimization, where the control vector is comprised of the amplitudes and relative phases of the inlet modes; the constraints are the prescribed total energy and that the flow evolution satisfies the full nonlinear compressible Navier-Stokes equations; the cost function is the average wall friction which, once maximized, corresponds to the earliest possible transition location. An ensemble variational (EnVar) technique is developed to solve the optimization problem. EnVar updates the estimate of the control vector at the end of each iteration using the gradient of the cost function, which is computed from the outcome of an ensemble of possible solutions. Two inflow conditions are computed, each corresponding to a different level of energy, and their spectra are contrasted: the lower energy case includes two normal acoustic waves and one oblique vorticity perturbation, whereas the higher energy condition consists of oblique acoustic and vorticity waves. The focus is placed on the former case, where the instability wave that initiates the rapid breakdown to turbulence is not present at the inlet plane. Instead, it appears at a downstream location after a series of nonlinear interactions that spur the fastest onset of turbulence. The results from the nonlinearly most potent inflow condition are also compared to other inlet disturbances that are selected solely based on linear theory, and which all yield relatively delayed transition onset.

physics.flu-dyn↗

arXiv2018

Perturbative expansions of the conformation tensor in viscoelastic flows

We consider the problem of formulating perturbative expansions of the conformation tensor, which is a positive-definite tensor representing polymer deformation in viscoelastic flows. The classical approach does not explicitly take into account that the perturbed tensor must remain positive definite---a fact that has important physical implications, e.g. extensions and compressions are represented similarly to within a negative sign, when physically the former are unbounded and the latter are bounded from below. Mathematically, the classical approach assumes that the underlying geometry is Euclidean, and this assumption is not satisfied by the manifold of positive definite tensors. We provide an alternative formulation that retains the conveniences of classical perturbation methods used for generating linear and weakly nonlinear expansions, but also provides a clear physical interpretation and a mathematical basis for analysis. The approach is based on treating a perturbation as a sequence of successively smaller deformations of the polymer. Each deformation is modelled explicitly using geodesics on the manifold of positive-definite tensors. Using geodesics, and associated geodesic distances, is the natural way to model perturbations to positive-definite tensors because it is consistent with the manifold geometry. Approximations of the geodesics can then be used to reduce the total deformation to a series expansion in the small perturbation limit. We illustrate our approach using direct numerical simulations of the nonlinear evolution of Tollmien-Schlichting waves.

physics.flu-dyn↗

arXiv2018

Geometric decomposition of the conformation tensor in viscoelastic turbulence

This work introduces a mathematical approach to analysing the polymer dynamics in turbulent viscoelastic flows that uses a new geometric decomposition of the conformation tensor, along with associated scalar measures of the polymer fluctuations. The approach circumvents an inherent difficulty in traditional Reynolds decompositions of the conformation tensor: the fluctuating tensor fields are not positive-definite and so do not retain the physical meaning of the tensor. The geometric decomposition of the conformation tensor yields both mean and fluctuating tensor fields that are positive-definite. The fluctuating tensor in the present decomposition has a clear physical interpretation as a polymer deformation relative to the mean configuration. Scalar measures of this fluctuating conformation tensor are developed based on the non-Euclidean geometry of the set of positive-definite tensors. Drag-reduced viscoelastic turbulent channel flow is then used an example case study. The conformation tensor field, obtained using direct numerical simulations, is analysed using the proposed framework.

physics.flu-dyn↗

arXiv2018

Scaling of energy amplification in viscoelastic channel and Couette flow

The linear amplification of disturbances is critical in setting up transition scenarios in viscoelastic channel and Couette flow, and may also play an important role when such flows are fully turbulent. As such, it is of interest to assess how this amplification, defined as the steady-state variance maintained under Gaussian white noise forcing, scales with the main nondimensional parameters: the Reynolds ($Re$) and Weissenberg ($Wi$) numbers. This scaling is derived analytically in the two limits of strong and weak elasticity for when the forcing is streamwise-constant. The latter is the relevant forcing for capturing the overall behaviour because it was previously shown to have the dominant contribution to amplification. The final expressions show that for weak elasticity the scaling retains a form similar to the well-known O($Re^3$) relationship with an added elastic correction. For strong elasticity, however, the scaling is O($Wi^3$) with a viscous correction. The key factor leading to such a mirroring in the scaling is the introduction of forcing in the polymer stress. The results demonstrate that energy amplification in a viscoelastic flow can be very sensitive to the model parameters even at low $Re$. They also suggest that energy amplification can be significantly increased by forcing the polymer stress, thereby opening up possibilities such as flow control using systematically designed polymer stress perturbations.

physics.flu-dyn↗

arXiv2016

Spectral analysis of the Navier-Stokes equations using the combination matrix

This work is a continuation of the analysis first presented in Cheung & Zaki (2014). In that study, the combination matrix was introduced as a means to tractably handle the nonlinear terms in the spectral domain. In this work, a different approach is discussed. Rather than analyze solutions to the energy equation, we examine the forced Navier-Stokes equations in spectral space and determine if direct solutions to the momentum equations can be found. This is done by using the combination matrix to rewrite the Navier-Stokes as a system of intersecting quadratic polynomials. Intrepreted geometrically, any solution to the Navier-Stokes can be represented as a the intersection of a multiple conic sections.

physics.flu-dyn↗